Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers

If rock beats scissors and scissors beat paper, onemight assume that rock beats paper too. But this is not the case for intransitive relationships thatmake up the famous rock-paper-scissors game. However, the sole presence of papermight prevent rock frombeating scissors, simply because paper beats rock. This is the blueprint for the rock-paper-scissors gamewith protection spillovers, which has recently been introduced as a newparadigm for biodiversity inwell-mixedmicrobial populations. Herewe study the game in structured populations, demonstrating that protection spillovers give rise to spatial patterns that are impossible to observe in the classical rock-paper-scissors game.We show that the spatiotemporal dynamics of the system is determined by the density of stable vortices, which may ultimately transform to frozen states, to propagating waves, or to target waves with reversed propagation direction, depending further on the degree and type of randomness in the interactions among the species. If vortices are rare, the fixation towaves and complex oscillatory solutions is likelier.Moreover, annealed randomness in interactions favors the emergence of target waves, while quenched randomness favors collective synchronization.Our results demonstrate that protection spilloversmay fundamentally change the dynamics of cyclic dominance in structured populations, and they outline the possibility of programming pattern formation inmicrobial populations.